All of the 4th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$2.00$ each for students, and the group paid $$42.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$19.50$ each for teachers and $$12.50$ each for students, and the group paid $$178.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+2y = 42}$ ${19.5x+12.5y = 178}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-19.5x-6y = -126}$ ${19.5x+12.5y = 178}$ Add the top and bottom equations together. $ 6.5y = 52 $ $ y = \dfrac{52}{6.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {6.5x+2y = 42}$ to find $x$ ${6.5x + 2}{(8)}{= 42}$ $6.5x+16 = 42$ $6.5x = 26$ $x = \dfrac{26}{6.5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {19.5x+12.5y = 178}$ and get the same answer for $x$ ${19.5x + 12.5}{(8)}{= 178}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.